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\begin{center}
\textbf{\LARGE Supplementary Material to }

\textbf{\LARGE "Estimation of Stochastic Volatility Models by Nonparametric\
Filtering"}

\bigskip

$%
\begin{tabular}{ccc}
\textsc{Shin Kanaya} &  & \textsc{Dennis Kristensen} \\ 
\textsc{University of Aarhus and CREATES} &  & \textsc{UCL, IFS and CREATES}%
\end{tabular}%
$
\end{center}

\bigskip

We here provide proofs of some technical results, including Lemma D.1 and
Theorem 7.1.\bigskip 

\noindent \textbf{Proof of }$\sup_{t\in \lbrack 0,T]}\left\vert
B_{t}\right\vert =o_{a.s.}(T^{1/2}\log T)$\textbf{\ as }$T\rightarrow \infty 
$\textbf{\ (Section 3.3). }By the reflection principle of the BM (e.g.,
Section 2.6 of Karatzas and Shreve, 1991), we have $\Pr \left[ \max_{t\leq
T}B_{t}>\bar{C}\right] =2-2\Phi (\bar{C}/\sqrt{T})$ for any $\bar{C}>0$, and 
$\Pr \left[ \min_{t\leq T}B_{t}<\underline{C}\right] =2\Phi (\underline{C}/%
\sqrt{T})$ for any $\underline{C}<0$, where $\Phi $ denotes the cumulative
distribution function of the standard normal. Then, for any $\bar{c}>0$,%
\begin{eqnarray*}
&&\sum\nolimits_{J=1}^{\infty }\Pr \left[ \max\nolimits_{t\leq J}B_{t}>\bar{c%
}J^{1/2}\left( \log J\right) \right] =2\sum\nolimits_{J=1}^{\infty }\left\{
1-\Phi \left( \bar{c}\log J\right) \right\} \\
&\leq &2\sum\nolimits_{J=1}^{\infty }\frac{1}{\bar{c}(\log J)}\exp \{-\bar{c}%
^{2}\left( \log J\right) ^{2}/2\}<\infty ,
\end{eqnarray*}%
where we have used the inequality $1-\Phi \left( x\right) \leq \left(
1/x\right) \exp \left\{ -x^{2}/2\right\} $ (Problem 9.22, Ch. 2, Karatzas
and Shreve, 1991). We can analogously show that for any $\underline{c}<0$,%
\begin{equation*}
\sum\nolimits_{J=1}^{\infty }\Pr \left[ \min\nolimits_{t\leq J}B_{t}<%
\underline{c}J^{1/2}\sqrt{\log J}\right] \leq 2\sum\nolimits_{J=1}^{\infty
}\left\{ 1-\Phi \left( \left\vert \bar{c}\right\vert \log J\right) \right\}
<\infty .
\end{equation*}%
These inequalities, together with the Borel-Cantelli lemma, imply that $%
\max_{t\leq J}\left\vert B_{t}\right\vert =o_{a.s.}(J^{1/2}\log J)$ for any
integer $J\geq 1$. For any arbitrary real number $T\geq 1$, we set $J=\lceil
T\rceil +1$, where $\lceil T\rceil $ denotes the integer part of $T$, and
then obtain%
\begin{equation*}
\max\nolimits_{t\leq T}\left\vert B_{t}\right\vert \leq \max\nolimits_{t\leq
\lceil T\rceil +1}\left\vert B_{t}\right\vert =o_{a.s.}(\left( \lceil
T\rceil +1\right) ^{1/2}\log \left( \lceil T\rceil +1\right)
)=o_{a.s.}\left( T^{1/2}\log T\right) .
\end{equation*}%
$\blacksquare $\bigskip

\noindent \textbf{Lemma D.1.} \textit{Suppose that a stochastic process }$%
\{\Gamma _{t}\}_{t\geq 0}$\textit{\ on a probability space }$\left( \Omega ,%
\mathfrak{F},P\right) $\textit{\ satisfies the condition:}%
\begin{equation}
E\left[ \left\vert \Gamma _{t}-\Gamma _{s}\right\vert ^{a}\right] \leq
C\left\vert t-s\right\vert ^{1+b},  \label{eq: mo}
\end{equation}%
\textit{for some positive constants }$a,$\textit{\ }$b$\textit{\ and }$C$%
\textit{\ each of which is independent of }$s$\textit{\ and }$t$\textit{.
Then, there exists a continuous modification }$\{\tilde{\Gamma}_{t}\}_{t\geq
0}$\textit{\ of }$\{\Gamma _{t}\}_{t\geq 0}$\textit{, which is a.s. H\"{o}%
lder continuous with exponent }$d$\textit{\ for every }$d\in \left(
0,b/a\right) $\textit{\ with a coefficient }$\vartheta :=\sum_{J=1}^{\infty
}J^{2+d}\left( 1/J!\right) ^{d}$\textit{:}%
\begin{equation}
\Pr \left[ \omega \in \Omega \left\vert \exists \bar{\Delta}\left( \omega
\right) \text{ s.t. }\sup_{\left\vert t-s\right\vert \in \left( 0,\bar{\Delta%
}\left( \omega \right) \right) ;\text{ }s,t\in \lbrack 0,\infty )}\frac{|%
\tilde{\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega
\right) |}{\left\vert t-s\right\vert ^{d}}\leq \vartheta \right. \right] =1.
\label{eq: as-holder}
\end{equation}

\begin{proof}
The following arguments proceed along the lines of the proof of Theorem
2.2.8 in Karatzas and Shreve (1991) where $s,t$ are supposed to take values
in some finite interval $\left[ 0,T\right] $ ($T=\bar{T}$ fixed). We first
prove the global H\"{o}lder property of the process on an enlarging
interval, i.e., $\left[ 0,T\right] $ where $T\rightarrow \infty $, and next
show that it actually holds over the infinite interval $[0,\infty )$. For
any $\varepsilon >0$, we have 
\begin{equation*}
\Pr \left[ \left\vert \Gamma _{t}-\Gamma _{s}\right\vert \geq \varepsilon %
\right] \leq \frac{E\left[ \left\vert \Gamma _{t}-\Gamma _{s}\right\vert ^{a}%
\right] }{\varepsilon ^{a}}\leq C\varepsilon ^{-a}\left\vert t-s\right\vert
^{1+b}
\end{equation*}%
by \v{C}eby\v{s}ev's inequality, and thus $\Gamma _{t}\overset{P}{%
\rightarrow }\Gamma _{s}$ as $s\rightarrow t$. Setting $t=km/m!$, $s=\left(
k-1\right) m/m!$ and $\varepsilon =\left( m/m!\right) ^{d}$, we obtain $\Pr %
\left[ \left\vert \Gamma _{km/m!}-\Gamma _{\left( k-1\right)
m/m!}\right\vert \geq \left( m/m!\right) ^{d}\right] \leq C\left(
m/m!\right) ^{\left( 1+b-ad\right) }$ and consequently, 
\begin{equation*}
\Pr \left[ \max_{1\leq k\leq m!}\left\vert \Gamma _{km/m!}-\Gamma _{\left(
k-1\right) m/m!}\right\vert \geq \left( m/m!\right) ^{d}\right] \leq
Cm^{\left( 1+b-ad\right) }\left/ \left( m!\right) ^{b-ad}\right. .
\end{equation*}%
By the fact that $\sum_{m=1}^{\infty }m^{\left( 1+b-ad\right) }/(m!)^{b-ad}$
exists and the Borel-Cantelli lemma, there exists a set $\Omega ^{\ast }\in 
\mathfrak{F}$ with $\Pr \left( \Omega ^{\ast }\right) =1$ such that 
\begin{equation}
\forall \omega \in \Omega ^{\ast },\exists m^{\ast }\left( \omega \right) ,\
\forall m\geq m^{\ast }\left( \omega \right) :\max_{1\leq k\leq
m!}\left\vert \Gamma _{km/m!}-\Gamma _{\left( k-1\right) m/m!}\right\vert
<\left( m/m!\right) ^{d}  \label{eq: bcl}
\end{equation}%
, where $m^{\ast }$ is a positive and integer-valued random variable. For
each integer $m\left( \geq 1\right) $ and any integer $l\geq m$, consider
the following sets, $E_{l}^{m}:=\left\{ km/l!\text{ }|\text{ }k=0,1,\dots
,l!\right\} $ and \ $E^{m}:=\tbigcup\nolimits_{l=1}^{\infty }E_{l}^{m}$. The
set $E_{l}^{m}$ consists of $\left( l!+1\right) $ points in $\left[ 0,m%
\right] $, while $E^{m}$ consists of infinitely many points in $\left[ 0,m%
\right] $.\ Note that $E^{m}$ is dense in $\left[ 0,m\right] $ for any $m$.\
Now fix $\omega \left( \in \Omega ^{\ast }\right) $ and $m\left( \geq
m^{\ast }\left( \omega \right) \right) $.\ We shall show that%
\begin{equation}
\forall l>m,\forall t,s\in E_{l}^{m}\text{ with}\ \left\vert t-s\right\vert
\in \left( 0,m/m!\right) :\left\vert \Gamma _{t}\left( \omega \right)
-\Gamma _{s}\left( \omega \right) \right\vert \leq
2\dsum\limits_{J=m+1}^{l}J^{2}\left[ J+1/\left( \left( J+1\right) !\right) %
\right] ^{d}.  \label{eq: tbs}
\end{equation}%
To show this, we use the inductive method. First, we prove that the claim is
true for $l=m+1$.\ For any $s,t\in E_{m+1}^{m}$ with $\left\vert
t-s\right\vert \in \left( 0,m/m!\right) $, there exist some $k_{1},k_{2}\in
\left\{ 0,1,\dots ,\left( m+1\right) !\right\} $ with $0\leq k_{2}-k_{1}\leq
m$ such that 
\begin{align*}
\left\vert \Gamma _{t}\left( \omega \right) -\Gamma _{s}\left( \omega
\right) \right\vert & \leq \left\vert \Gamma _{k_{1}\left( m+1\right)
/\left( m+1\right) !}\left( \omega \right) -\Gamma _{(k_{1}+1)\left(
m+1\right) /\left( m+1\right) !}\left( \omega \right) \right\vert  \\
& +\left\vert \Gamma _{(k_{1}+1)\left( m+1\right) /\left( m+1\right)
!}\left( \omega \right) -\Gamma _{(k_{1}+2)\left( m+1\right) /\left(
m+1\right) !}\left( \omega \right) \right\vert  \\
& +\dots +\left\vert \Gamma _{(k_{2}-1)\left( m+1\right) /\left( m+1\right)
!}\left( \omega \right) -\Gamma _{k_{2}\left( m+1\right) /\left( m+1\right)
!}\left( \omega \right) \right\vert .
\end{align*}%
Each term on the right-hand side is bounded by $m\left[ \left( m+2\right)
/\left( m+2\right) !\right] ^{d}$, which is implied by the fact $%
E_{m+1}^{m}\subset E_{m+2}^{m+2}$ and the inequality eq. (\ref{eq: bcl}).
Thus, by the triangle inequalities, we have $\left\vert \Gamma _{t}\left(
\omega \right) -\Gamma _{s}\left( \omega \right) \right\vert \leq m^{2}\left[
\left( m+2\right) /\left( m+2\right) !\right] ^{d}$. Second, suppose that
eq. (\ref{eq: tbs}) is valid for $l=m+1,\dots ,L-1$.\ For $s<t,$ ($s,t\in
E_{L}^{m}$) with $\left\vert t-s\right\vert \in \left( 0,m/m!\right) $,
consider the numbers $s_{1}:=\min \left\{ u\in E_{L-1}^{m}:u\geq s\right\} $
and $t_{1}:=\max \left\{ u\in E_{L-1}^{m}:u\leq t\right\} $, and notice that 
$s_{1},t_{1}\in E_{L-1}^{m}\subset E_{L}^{m}\subset E_{L+1}^{L+1}$; $s,t\in
E_{L}^{m}\subset E_{L+1}^{L+1}$; $s_{1}-s<m/\left( L-1\right) !$; and $%
t-t_{1}<m/\left( L-1\right) !$.\ By the inequality eq. (\ref{eq: bcl}) with $%
m=L+1$, $\left\vert \Gamma _{s_{1}}\left( \omega \right) -\Gamma _{s}\left(
\omega \right) \right\vert \leq mL\left( \left( L+1\right) /\left(
L+1\right) !\right) ^{d}$ and $\left\vert \Gamma _{t}\left( \omega \right)
-\Gamma _{t_{1}}\left( \omega \right) \right\vert \leq mL\left( \left(
L+1\right) /\left( L+1\right) !\right) ^{d}$. There are two possible
relationships among $s,t,s_{1}$ and $t_{1}$: (i) if$\ \left\vert
t-s\right\vert \geq m/\left( L-1\right) !$, it holds that $s\leq s_{1}\leq
t_{1}\leq t$ (with at least one inequality strict); (ii) if $\left\vert
t-s\right\vert <m/\left( L-1\right) !$, either of $\left\vert t-s\right\vert
<\left\vert s_{1}-t_{1}\right\vert =m/\left( L-1\right) !$ or $\left\vert
s_{1}-t_{1}\right\vert =0$.\ Thus, we have $\left\vert
t_{1}-s_{1}\right\vert \leq \max \left\{ m/\left( L-1\right) !,\left\vert
t-s\right\vert \right\} \leq m/m!$, and use the induction assumption (\ref%
{eq: tbs}) with $l=L-1$:%
\begin{equation}
\left\vert \Gamma _{t_{1}}\left( \omega \right) -\Gamma _{s_{1}}\left(
\omega \right) \right\vert \leq 2\sum_{J=m+1}^{L-1}J^{2}\left[ \dfrac{J+1}{%
\left( J+1\right) !}\right] ^{d}.  \label{eq: approx-point}
\end{equation}%
Therefore, 
\begin{equation*}
\left\vert \Gamma _{t}\left( \omega \right) -\Gamma _{s}\left( \omega
\right) \right\vert \leq 2mL\left( \left( L+1\right) /\left( L+1\right)
!\right) ^{d}+2\sum_{J=m+1}^{L-1}J^{2}\left[ \dfrac{J+1}{\left( J+1\right) !}%
\right] ^{d}<2\sum_{J=m+1}^{L}J^{2}\left[ \dfrac{J+1}{\left( J+1\right) !}%
\right] ^{d}.
\end{equation*}%
We have shown eq. (\ref{eq: tbs}) for any $l\left( >m\right) $, as desired.

We can now show that $\left\{ \Gamma _{t}\left( \omega \right) \text{ }|%
\text{ }t\in E^{m}\right\} $ is uniformly H\"{o}lder in $t$ for $\forall
\omega \in \Omega ^{\ast }$ for any $m$.$\ $Consider any numbers $s,t\in
E^{m}$ with $m\geq m^{\ast }\left( =m^{\ast }\left( \omega \right) \right) $
and$\ \left\vert t-s\right\vert <\bar{\Delta}\left( \omega \right) \equiv
m^{\ast }/m^{\ast }!$. Note that $E^{m}\subseteq E^{m^{\prime }}$ for $m\leq
m^{\prime }$. We can pick some $m^{\prime }\left( \geq m\right) $ such that $%
s,t\in E^{m^{\prime }}$ with $\left( m^{\prime }+2\right) /\left( m^{\prime
}+2\right) !\leq t-s<\left( m^{\prime }+1\right) /\left( m^{\prime
}+1\right) !$. Then, by eq. (\ref{eq: tbs}), we obtain 
\begin{equation*}
\left\vert \Gamma _{t}\left( \omega \right) -\Gamma _{s}\left( \omega
\right) \right\vert \leq 2\sum_{J=m^{\prime }+1}^{\infty }J^{2}\left[ \dfrac{%
J+1}{\left( J+1\right) !}\right] ^{d}\leq \left[ \left( m^{\prime }+2\right)
/\left( m^{\prime }+2\right) !\right] ^{d}\times \sum_{J=1}^{\infty
}J^{2+d}\left( 1/J!\right) ^{d}
\end{equation*}%
and thus, $\left. \left\vert \Gamma _{t}\left( \omega \right) -\Gamma
_{s}\left( \omega \right) \right\vert \right/ \left\vert t-s\right\vert
^{d}\leq c_{h}$ where \ $\vartheta :=\sum_{J=1}^{\infty }J^{2+d}\left(
1/J!\right) ^{d}$. Note that the existence of $\vartheta $ can be checked by
d'Alembert's criterion for any $d\left( >0\right) $.

We define $\{\tilde{\Gamma}_{t}\}_{t\geq 0}$ as follows. For $\omega \notin
\Omega ^{\ast }$, set $\tilde{\Gamma}_{t}\left( \omega \right) =0$, $t\in %
\left[ 0,m\right] $. For $\omega \in \Omega ^{\ast }$ and $t\in E^{m}$, set $%
\tilde{\Gamma}_{t}\left( \omega \right) =\Gamma _{t}\left( \omega \right) $.
\ For $\omega \in \Omega ^{\ast }$ and $t\in \left[ 0,m\right] \cap \left(
E^{m}\right) ^{c}$, choose a sequence $\left\{ s_{n}\right\} _{n=1}^{\infty }
$with $s_{n}\left( \in E^{m}\right) \rightarrow t$; by the uniform
continuity and the fact that $s_{n}$ is Cauchy, $\left\{ \Gamma
_{s_{n}}\left( \omega \right) \right\} _{n=1}^{\infty }$ is also Cauchy,
whose limit depends of $t$ but not on the particular sequence $\left\{
s_{n}\right\} $; and thus let $\tilde{\Gamma}_{t}\left( \omega \right)
=\lim_{s_{n}\rightarrow t}\Gamma _{s_{n}}\left( \omega \right) $.\ Thus, the
resulting process $\{\tilde{\Gamma}_{t}\}_{t\in \left[ 0,m\right] }$ is
continuous, and is also uniformly H\"{o}lder in $t\in \left[ 0,m\right] $.
We will show $\{\tilde{\Gamma}_{t}\}$ is a modification of $\{\Gamma _{t}\}$%
: observe that for $t\in E^{m},\ \tilde{\Gamma}_{t}=\Gamma _{t}$ a.s.; for $%
t\in \left[ 0,m\right] \cap \left( E^{m}\right) ^{c}$ and $\left\{
s_{n}\right\} $ with $s_{n}\left( \in E^{m}\right) \rightarrow t,$ we have $%
\Gamma _{s_{n}}\rightarrow \Gamma _{t}$ in probability (by eq. (\ref{eq: mo}%
)) and $\Gamma _{s_{n}}\rightarrow \tilde{\Gamma}_{t}$ a.s., which implies $%
\tilde{\Gamma}_{t}=\Gamma _{t}$ a.s.

Let $m=\left[ T\right] +1$ with $\left[ T\right] $ denoting the largest
integer less than or equal to $T$. Now, we have proved that for any $\omega
\in \Omega ^{\ast }$, there exist some $m^{\ast }\left( \omega \right) $ and 
$\bar{\Delta}\left( \omega \right) \left( \equiv m^{\ast }/m^{\ast }!\right) 
$ such that $\forall m\geq m^{\ast }\left( \omega \right) $ 
\begin{equation*}
\sup_{\substack{ \left\vert t-s\right\vert \in \left( 0,\Bar{\Delta}\left(
\omega \right) \right)  \\ t,s\in \left[ 0,T\right] }}\left. \left\vert 
\tilde{\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega
\right) \right\vert \right/ \left\vert t-s\right\vert ^{d}\leq \sup
_{\substack{ \left\vert t-s\right\vert \in \left( 0,\Bar{\Delta}\left(
\omega \right) \right)  \\ t,s\in \left[ 0,m\right] }}\left. \left\vert 
\tilde{\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega
\right) \right\vert \right/ \left\vert t-s\right\vert ^{d}\leq \vartheta ,
\end{equation*}%
which implies that $\Pr \left( \Omega _{1}\right) =1$, where 
\begin{equation}
\Omega _{1}:=\left\{ \exists \bar{\Delta}\left( \omega \right) ,\text{ }%
\exists T^{\ast },\text{ }\forall T\left( \geq T^{\ast }\right) ,\text{ }%
\sup_{\left\vert t-s\right\vert \in \left( 0,\bar{\Delta}\left( \omega
\right) \right) ;\text{ \ }s,t\in \left[ 0,T\right] }\tfrac{\left\vert 
\tilde{\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega
\right) \right\vert }{\left\vert t-s\right\vert ^{d}}\leq \vartheta \right\}
.  \label{eq: prob-1}
\end{equation}%
Note that 
\begin{align}
\Omega _{1}& \subset \left\{ \exists \bar{\Delta}\left( \omega \right) ,%
\text{ }\exists T^{\ast },\text{\ }\sup_{\left\vert t-s\right\vert \in
\left( 0,\bar{\Delta}\left( \omega \right) \right) ;\text{ }s,t\in \left[
0,T^{\ast }\right] }\tfrac{\left\vert \tilde{\Gamma}_{t}\left( \omega
\right) -\tilde{\Gamma}_{s}\left( \omega \right) \right\vert }{\left\vert
t-s\right\vert ^{d}}\leq \vartheta \right\}   \notag \\
& =\left\{ \exists \bar{\Delta}\left( \omega \right) ,\text{ }\exists
T^{\ast },\text{ }\forall T\left( \leq T_{\ast }\right) ,\text{\ }%
\sup_{\left\vert t-s\right\vert \in \left( 0,\bar{\Delta}\left( \omega
\right) \right) ;\text{ }s,t\in \left[ 0,T\right] }\tfrac{\left\vert \tilde{%
\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega \right)
\right\vert }{\left\vert t-s\right\vert ^{d}}\leq \vartheta \right\}
=:\Omega _{2}.  \label{eq: prob-inc}
\end{align}%
Since $\Pr \left( \Omega _{1}\right) =1$, we then have $\Pr \left( \Omega
_{2}\right) =1$. For any events $E,F\in \mathfrak{F},$ we have the
inequality: $\Pr \left( E\cap F\right) \geq \Pr \left( E\right) +\Pr \left(
F\right) -1$.\ With $E=\Omega _{1}$ and $F=\Omega _{2}$, we obtain $\Pr
\left( \Omega _{1}\cap \Omega _{2}\right) =1$, which, together with 
\begin{equation*}
\Omega _{1}\cap \Omega _{2}=\left\{ \omega \in \Omega \left\vert \exists 
\bar{\Delta}\left( \omega \right) ,\text{ }\forall T,\text{ }%
\sup_{\left\vert t-s\right\vert \in \left( 0,\bar{\Delta}\left( \omega
\right) \right) ;\text{ }s,t\in \left[ 0,T\right] }\tfrac{\left\vert \tilde{%
\Gamma}_{t}\left( \omega \right) -\tilde{\Gamma}_{s}\left( \omega \right)
\right\vert }{\left\vert t-s\right\vert ^{d}}\leq \vartheta \right. \right\}
,
\end{equation*}%
implies the desired result, eq. (\ref{eq: as-holder}).
\end{proof}

\bigskip

\begin{proof}[Proof of Theorem 7.1]
Let 
\begin{eqnarray*}
U_{1}\left( j\right)  &:&=2\partial _{\theta _{1}}\alpha (\sigma _{\tau
_{j}}^{2};\theta _{1}^{\ast })\left[ \alpha (\sigma _{\tau _{j}}^{2};\theta
_{1}^{\ast })\delta -(\sigma _{\tau _{j+1}}^{2}-\sigma _{\tau _{j}}^{2})%
\right] ; \\
U_{2}\left( j\right)  &:&=2\partial _{\theta _{2}}\beta ^{2}(\sigma _{\tau
_{j}}^{2};\theta _{2}^{\ast })[\beta ^{2}(\sigma _{\tau _{j}}^{2};\theta
_{2}^{\ast })\delta -(\sigma _{\tau _{j+1}}^{2}-\sigma _{\tau
_{j}}^{2})^{2}].
\end{eqnarray*}%
Then, we can then write 
\begin{eqnarray*}
\hat{S}_{k}(\theta _{k}^{\ast },\sigma ^{2})
&=&T^{-1}\sum\nolimits_{j=1}^{N-1}U_{k}\left( j\right) \text{ \ and \ } \\
H_{k}^{\ast -1}E\left[ \hat{S}_{k}(\theta _{k}^{\ast },\sigma ^{2})\hat{S}%
_{k}(\theta _{k}^{\ast },\sigma ^{2})^{\bigstar }\right] H_{k}^{\ast -1} &=&%
\mathcal{B}_{\theta _{k}}\mathcal{B}_{\theta _{k}}^{^{\bigstar }}+\mathcal{V}%
_{\theta _{k}}+\mathcal{C}_{\theta _{k}},
\end{eqnarray*}%
where%
\begin{eqnarray*}
\mathcal{B}_{\theta _{k}} &:&=-H_{k}^{\ast -1}E\left[ \hat{S}_{k}\left(
\theta _{k}^{\ast },\sigma ^{2}\right) \right] ; \\
\mathcal{V}_{\theta _{k}} &:&=H_{k}^{\ast -1}\left(
T^{-2}\sum\nolimits_{j=1}^{N-1}E\left[ \left( U_{k}\left( j\right) -E\left[
U_{k}\left( j\right) \right] \right) \left( U_{k}\left( j\right) -E\left[
U_{k}\left( j\right) \right] \right) ^{\bigstar }\right] \right) H_{k}^{\ast
-1}; \\
\mathcal{C}_{\theta _{k}} &:&=H_{k}^{\ast -1}\left(
T^{-2}\sum\nolimits_{1\leq i\neq j\leq N-1}E\left[ \left( U_{k}\left(
i\right) -E\left[ U_{k}\left( i\right) \right] \right) \left( U_{k}\left(
j\right) -E\left[ U_{k}\left( j\right) \right] \right) ^{\bigstar }\right]
\right) H_{k}^{\ast -1}.
\end{eqnarray*}%
We below provide the proof for part (i) ($k=1$) only. Part (ii) ($k=2$) can
be proved in the same way, and its proof is omitted. Let $\mathcal{L}$ be
the differential operator defined by $\mathcal{L}f\left( x\right) =f^{\prime
}\left( x\right) \alpha \left( x\right) +f^{\prime \prime }\left( x\right)
\beta ^{2}\left( x\right) /2$ for any twice differentiable function $f$. We
first consider the expression of $\mathcal{B}_{\theta _{1}}$:%
\begin{eqnarray}
E\left[ U_{k}\left( j\right) \right]  &=&-\delta ^{2}E\left[ \partial
_{\theta _{1}}\alpha (\sigma _{u}^{2};\theta _{1}^{\ast })\mathcal{L}\alpha
\left( \sigma _{u}^{2}\right) \right] +2\int_{\tau _{j}}^{\tau
_{j+1}}\int_{\tau _{j}}^{s}\int_{\tau _{j}}^{u}E\left[ \mathcal{L}\partial
_{\theta _{1}}\alpha (\sigma _{v}^{2};\theta _{1}^{\ast })\mathcal{L}\alpha
\left( \sigma _{u}^{2}\right) \right] dvduds  \notag \\
&=&-\delta ^{2}E\left[ \partial _{\theta _{1}}\alpha (\sigma _{t}^{2};\theta
_{1}^{\ast })\mathcal{L}\alpha \left( \sigma _{t}^{2}\right) \right] \left[
1+O\left( \delta \right) \right] ,  \label{eq: biasU}
\end{eqnarray}%
uniformly over $j$, where we have applied the martingale property of
stochastic integrals and Ito's lemma to $\alpha \left( \sigma
_{s}^{2}\right) -\alpha (\sigma _{\tau _{j}}^{2})$ and $\partial _{\theta
_{1}}\alpha (\sigma _{u}^{2};\theta _{1}^{\ast })-\partial _{\theta
_{1}}\alpha (\sigma _{\tau _{j}}^{2};\theta _{1}^{\ast })$, and the last
equality holds since 
\begin{equation*}
E\left[ \mathcal{L}\partial _{\theta _{1}}\alpha (\sigma _{v}^{2};\theta
_{1}^{\ast })\mathcal{L}\alpha \left( \sigma _{u}^{2}\right) \right] \leq
\left\{ E\left[ \left\vert \mathcal{L}\partial _{\theta _{1}}\alpha (\sigma
_{v}^{2};\theta _{1}^{\ast })\right\vert ^{2}\right] E\left[ \left\vert 
\mathcal{L}\alpha \left( \sigma _{u}^{2}\right) \right\vert ^{2}\right]
\right\} ^{1/2}\leq E\left[ \left\vert \psi \left( \sigma _{t}^{2}\right)
\right\vert ^{4}\right] =O\left( 1\right) ,
\end{equation*}%
uniformly over any $u$ and $v$, which follows from the moment conditions in
Assumption C-SDR. Now, the above definition of $\mathcal{B}_{\theta _{k}}$
and eq. (\ref{eq: biasU}) implies eq. (7.3) of the main text. To find the
expression of $\mathcal{V}_{\theta _{k}}$, first note that%
\begin{eqnarray*}
&&E\left[ U_{1}\left( j\right) U_{1}\left( j\right) ^{\bigstar }\right] =4E%
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\biggl[%
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\partial _{\theta _{1}}\alpha (\sigma _{t}^{2};\theta _{1}^{\ast })\partial
_{\theta _{1}}\alpha (\sigma _{t}^{2};\theta _{1}^{\ast })^{\bigstar } \\
&&\times \left\{ \alpha ^{2}(\sigma _{\tau _{j}}^{2})\delta ^{2}-2\alpha
(\sigma _{\tau _{j}}^{2})\delta \left( \int_{\tau _{j}}^{\tau _{j+1}}\alpha
\left( \sigma _{s}^{2}\right) ds+\int_{\tau _{j}}^{\tau _{j+1}}\beta \left(
\sigma _{s}^{2}\right) dZ_{s}\right) \right.  \\
&&+2\int_{\tau _{j}}^{\tau _{j+1}}\left( \int_{\tau _{j}}^{s}\alpha (\sigma
_{u}^{2})du+\int_{\tau _{j}}^{s}\beta (\sigma _{u}^{2})dZ_{u}\right) \alpha
(\sigma _{s}^{2})ds \\
&&\left. +2\int_{\tau _{j}}^{\tau _{j+1}}\left( \int_{\tau _{j}}^{s}\alpha
(\sigma _{u}^{2})du+\int_{\tau _{j}}^{s}\beta (\sigma _{u}^{2})dZ_{u}\right)
\beta (\sigma _{s}^{2})dZ_{s}+\int_{\tau _{j}}^{\tau _{j+1}}\beta
^{2}(\sigma _{s}^{2})ds\right\} 
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\\
&=&\delta \underset{=\Omega _{1}^{\ast }}{\underbrace{E\left[ \partial
_{\theta _{1}}\alpha (\sigma _{t}^{2};\theta _{1}^{\ast })\partial _{\theta
_{1}}\alpha (\sigma _{t}^{2};\theta _{1}^{\ast })^{\bigstar }\beta
^{2}\left( \sigma _{t}^{2}\right) \right] }}\left[ 1+O\left( \delta \right) %
\right] ,\text{ uniformly over }j
\end{eqnarray*}%
where Ito's lemma is applied to $(\sigma _{\tau _{j+1}}^{2}-\sigma _{\tau
_{j}}^{2})^{2}$ in the first equality; and the second equality follows from
arguments similar to those in deriving eq. (\ref{eq: biasU}). By the
definition of $\mathcal{V}_{\theta _{k}}$ and the result that $E\left[
U_{k}\left( j\right) \right] =O\left( \delta ^{2}\right) $ uniformly over $j$%
,%
\begin{equation*}
\mathcal{V}_{\theta _{k}}=H_{k}^{\ast -1}\left(
T^{-2}\sum\nolimits_{j=1}^{N-1}\left[ \delta \Omega _{1}^{\ast }\left[
1+o\left( 1\right) \right] -O\left( \delta ^{4}\right) \right] \right)
H_{k}^{\ast -1}=T^{-1}H_{k}^{\ast -1}\Omega _{1}^{\ast }H_{k}^{\ast -1}\left[
1+O\left( \delta \right) \right] ,
\end{equation*}%
as claimed. To find the expression of $\mathcal{C}_{\theta _{k}}$, we write 
\begin{equation*}
\partial _{\theta _{1}}\alpha \left( \sigma _{\tau _{j}}^{2};\theta
_{1}^{\ast }\right) [\alpha (\sigma _{\tau _{j}}^{2};\theta _{1}^{\ast
})\delta -(\sigma _{\tau _{j+1}}^{2}-\sigma _{\tau _{j}}^{2})]=:\Upsilon
_{1}\left( j\right) +\Upsilon _{2}\left( j\right) ,
\end{equation*}%
where $\Upsilon _{1}\left( j\right) :=-\partial _{\theta _{1}}\alpha \left(
\sigma _{\tau _{j}}^{2};\theta _{1}^{\ast }\right) \int_{\tau _{j}}^{\tau
_{j+1}}\int_{\tau _{j}}^{s}\mathcal{L}\alpha \left( \sigma _{u}^{2}\right)
duds$ and%
\begin{equation*}
\Upsilon _{2}\left( j\right) :=\partial _{\theta _{1}}\alpha \left( \sigma
_{\tau _{j}}^{2};\theta _{1}^{\ast }\right) \left\{ \int_{\tau _{j}}^{\tau
_{j+1}}\int_{\tau _{j}}^{s}\alpha ^{\prime }\left( \sigma _{u}^{2}\right)
\beta \left( \sigma _{u}^{2}\right) dZ_{u}ds-\int_{\tau _{j}}^{\tau
_{j+1}}\beta \left( \sigma _{s}^{2}\right) dZ_{s}\right\} .
\end{equation*}%
Then, by the martingale property of stochastic integrals, Fubini's theorem
and the conditions in (C-SDR), $E\left[ \Upsilon _{1}\left( i\right)
\Upsilon _{2}\left( j\right) ^{\bigstar }\right] =0$ and $E\left[ \Upsilon
_{2}\left( i\right) \Upsilon _{2}\left( j\right) ^{\bigstar }\right] =0$ for 
$i\neq j$. Given the moment conditions in (C-SDR), we can show that $E\left[
\Upsilon _{1}\left( i\right) \Upsilon _{1}\left( j\right) ^{\bigstar }\right]
=O\left( \delta ^{4}\right) \ $uniformly over any $i\neq j$, by using
arguments analogous to those for $\mathcal{B}_{\theta _{k}}$ and $\mathcal{V}%
_{\theta _{k}}$. This, together with eq. (\ref{eq: biasU}),%
\begin{equation*}
E\left[ \left( U_{k}\left( i\right) -E\left[ U_{k}\left( i\right) \right]
\right) \left( U_{k}\left( j\right) -E\left[ U_{k}\left( j\right) \right]
\right) ^{\bigstar }\right] =E[\Upsilon _{1}\left( i\right) \Upsilon
_{1}\left( j\right) ^{\bigstar }]-E\left[ U_{k}\left( i\right) \right] E%
\left[ U_{k}\left( j\right) \right] ^{\bigstar }=O\left( \delta ^{4}\right) ,
\end{equation*}%
which, together with the definition of $\mathcal{C}_{\theta _{k}}$, implies
that $\mathcal{C}_{\theta _{k}}=O\left( \delta ^{2}\right) $. This completes
the proof.
\end{proof}

\bigskip 

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Karatzas, I. and S.E. Shreve (1991) \textit{Brownian Motion and Stochastic
Calculus}. Second edition, Springer.

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\end{document}
